Abstract
In this paper we prove an integral representation formula for the inverse Fueter mapping theorem for monogenic functions defined on axially symmetric open sets U ⊆ ℝn+1, i.e. on open sets U invariant under the action of SO(n), where n is an odd number. Every monogenic function on such an open set U can be written as a series of axially monogenic functions of degree k, i.e. functions of type \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} _k (x) = \left[ {A\left( {x_{0,\rho } } \right) + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } {\rm B}\left( {x_{0,\rho } } \right)} \right]\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )\), where A(x 0, ρ) and B(x 0, ρ) satisfy a suitable Vekua-type system and \(\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )\) is a homogeneous monogenic polynomial of degree k. The Fueter mapping theorem says that given a holomorphic function f of a paravector variable defined on U, then the function \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} _k (x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )\) given by $$\Delta ^{k + \tfrac{{n - 1}} {2}} \left( {f(x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )} \right) = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} (x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$$ is a monogenic function. The aim of this paper is to invert the Fueter mapping theorem by determining a holomorphic function f of a paravector variable in terms of \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} _k (x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )\). This result allows one to invert the Fueter mapping theorem for any monogenic function defined on an axially symmetric open set.
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